Theorem subrgmcl

Description: A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014) (Proof shortened by AV, 30-Mar-2025)

	Ref	Expression
Hypothesis	subrgmcl.p	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
Assertion	subrgmcl	$$⊢\left(A\in \mathrm{SubRing}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{·}Y\in A$$

Proof

Step	Hyp	Ref	Expression
1		subrgmcl.p	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
2		subrgsubrng	$$⊢A\in \mathrm{SubRing}\left(R\right)\to A\in \mathrm{SubRng}\left(R\right)$$
3	1	subrngmcl	$$⊢\left(A\in \mathrm{SubRng}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{·}Y\in A$$
4	2 3	syl3an1	$$⊢\left(A\in \mathrm{SubRing}\left(R\right)\wedge X\in A\wedge Y\in A\right)\to X\underset{\dot{}}{·}Y\in A$$